On the complexity of topological conjugacy of Toeplitz subshifts

Sabok, Marcin; Tsankov, Todor

doi:10.1007/s11856-017-1541-8

Cited by 7 publications

(5 citation statements)

References 26 publications

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“…The bireducibility type of the conjugacy relation of various classes of dynamical systems and group actions has been extensively studied. For many classes of topological or measurable dynamical systems, the conjugacy relations is known or conjectured to be quite complicated from the perspective of bireducibility [39,40,62,102]. As an example, the conjugacy relation of elements of Homeo 0 (R) is bireducible to the isomorphism relation of all countable (but not necessarily locally finite) graphs, which is analytic but not Borel (see [48,Theorem 4.9]), and the conjugacy of homeomorphisms of the plane is strictly more complicated by a result of Hjorth [48,Theorem 4.17].…”

Section: Topology Of the Space Of Equivalence Classes Of Preordersmentioning

confidence: 99%

Locally moving groups acting on the line and $\mathbb{R}$-focal actions

Brum,

Bon,

Rivas

et al. 2021

Preprint

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We prove various results that, given a sufficiently rich subgroup G of the group of homeomorphisms on the real line, describe the structure of the other possible actions of G on the real line, and address under which conditions such actions must be semi-conjugate to the natural defining action of G. The main assumption is that G should be locally moving, meaning that for every open interval the subgroup of elements whose support is contained in such interval acts on it without fixed points. One example (among many others) is given by Thompson's group F . A first rigidity result implies that if G is a locally moving group, every faithful minimal action of G on the real line by C 1 diffeomorphisms is conjugate to its standard action. It turns out that the situation is much wilder when considering actions by homeomorphisms: for a large class of groups, including Thompson's group F , we describe uncountably many conjugacy classes of minimal faithful actions by homeomorphisms on the real line. To gain insight on such exotic actions, we introduce and develop the notion of R-focal action, a class of actions on the real line that can be encoded by certain actions by homeomorphisms on planar real trees fixing an end. Under a suitable finite generation condition on a locally moving group G, we prove that every minimal faithful action of G on the line is either conjugate to the standard action, or it is R-focal and the action on the associated real tree factors via a horofunction onto the standard action of G on the line. This establishes a tight relation between all minimal actions of G on the line and its standard action. Among the various applications of this result, we show that for a large class of locally moving groups, the standard action is locally rigid, in the sense that every sufficiently small perturbation in the compact-open topology gives a semi-conjugate action. This is based on an analysis of the space of harmonic actions on the line for such groups. Along the way we introduce and study several concrete examples.

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Section: Topology Of the Space Of Equivalence Classes Of Preordersmentioning

confidence: 99%

Locally moving groups acting on the line and $\mathbb{R}$-focal actions

Brum,

Bon,

Rivas

et al. 2021

Preprint

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“…A subshift X of n Γ is Toeplitz if it is the closure of the Γ-orbit of some Toeplitz function x ∈ n Γ . Toeplitz subshifts are always minimal [31] and, as explained in [44], the set ToepSubsh(n Γ ) of all free Toeplitz subshifts of n Γ is a Borel subset of FMSubsh(n Γ ). We let NAToepSubsh(n Z 2 ) denote the Borel subset of ToepSubsh(n Z 2 ) consisting of all free Toeplitz subshifts X ⊆ n Z 2 with the property that the topological full group [[Z 2 X]] is nonamenable (equivalently, the commutator subgroup [[Z 2 X]] ′ is nonamenable).…”

Section: Isomorphism On the Space Of Finitely Generated Simple Noname...mentioning

confidence: 99%

Dynamical alternating groups, stability, property Gamma, and inner amenability

Kerr¹,

Tucker-Drob²

2019

Preprint

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We prove that the alternating group of a topologically free action of a countably infinite group Γ on the Cantor set has the property that all of its ℓ 2 -Betti numbers vanish and, in the case that Γ is amenable, is stable in the sense of Jones and Schmidt and has property Gamma (and in particular is inner amenable). We show moreover in the realm of amenable Γ that there are large classes of such alternating groups which are simple, finitely generated, and nonamenable, and in many cases also C * -simple. Among the tools used in constructing some of these classes is a topological version of Austin's result on the invariance of measure entropy under bounded orbit equivalence.

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“…Conjugacy of odometers is smooth due to Buescu and Stewart [2]. The complexity of conjugacy of Toeplitz subshifts was treated several times—by Thomas, Sabok, and Tsankov, and by Kaya [13, 26, 28]. Conjugacy of two-sided subshifts is Borel bireducible to the universal countable Borel equivalence relation due to Clemens [6].…”

Section: Introductionmentioning

confidence: 99%

Classification of One Dimensional Dynamical Systems by Countable Structures

Bruin

Vejnar

2022

J. symb. log.

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We study the complexity of the classification problem of conjugacy on dynamical systems on some compact metrizable spaces. Especially we prove that the conjugacy equivalence relation of interval dynamical systems is Borel bireducible to isomorphism equivalence relation of countable graphs. This solves a special case of Hjorth’s conjecture which states that every orbit equivalence relation induced by a continuous action of the group of all homeomorphisms of the closed unit interval is classifiable by countable structures. We also prove that conjugacy equivalence relation of Hilbert cube homeomorphisms is Borel bireducible to the universal orbit equivalence relation.

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On the complexity of topological conjugacy of Toeplitz subshifts

Cited by 7 publications

References 26 publications

Locally moving groups acting on the line and $\mathbb{R}$-focal actions

Locally moving groups acting on the line and $\mathbb{R}$-focal actions

Dynamical alternating groups, stability, property Gamma, and inner amenability

Classification of One Dimensional Dynamical Systems by Countable Structures

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